Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 39\cdot 79 + 43\cdot 79^{2} + 51\cdot 79^{3} + 16\cdot 79^{4} + 67\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 9\cdot 79 + 39\cdot 79^{2} + 34\cdot 79^{3} + 38\cdot 79^{4} + 36\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 75\cdot 79 + 70\cdot 79^{2} + 46\cdot 79^{3} + 75\cdot 79^{4} + 51\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 15\cdot 79 + 31\cdot 79^{2} + 65\cdot 79^{3} + 68\cdot 79^{4} + 15\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 46 + 63\cdot 79 + 47\cdot 79^{2} + 13\cdot 79^{3} + 10\cdot 79^{4} + 63\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 + 3\cdot 79 + 8\cdot 79^{2} + 32\cdot 79^{3} + 3\cdot 79^{4} + 27\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 66 + 69\cdot 79 + 39\cdot 79^{2} + 44\cdot 79^{3} + 40\cdot 79^{4} + 42\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 39\cdot 79 + 35\cdot 79^{2} + 27\cdot 79^{3} + 62\cdot 79^{4} + 11\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
